A Scaling Approach to Elliptic Theory for Geometrically-Natural Differential Operators with Sobolev-Type Coefficients

Michael Holst; David Maxwell; Gantumur Tsogtgerel

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arXiv · 2306.15842

A Scaling Approach to Elliptic Theory for Geometrically-Natural Differential Operators with Sobolev-Type Coefficients

Michael Holst·David Maxwell·Gantumur Tsogtgerel

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Abstract

We develop local elliptic regularity for operators having coefficients in a range of Sobolev-type function spaces (Bessel potential, Sobolev-Slobodeckij, Triebel-Lizorkin, Besov) where the coefficients have a regularity structure typical of operators in geometric analysis. The proofs rely on a nonstandard technique using rescaling estimates and apply to operators having coefficients with low regularity. For each class of function space for an operator's coefficients, we exhibit a natural associated range of function spaces of the same type for the domain of the operator and we provide regularity inference along with interior estimates. Additionally, we present a unified set of multiplication results for the function spaces we consider.

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Michael Holst, David Maxwell, Gantumur Tsogtgerel. 2023-06-28. A Scaling Approach to Elliptic Theory for Geometrically-Natural Differential Operators with Sobolev-Type Coefficients. https://arxiv.org/abs/2306.15842

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